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Topological equivalence between a 3D object and the reconstruction of its digital image
 IEEE Trans. Pattern Anal. Mach. Intell
, 2007
"... Digitization is not as easy as it looks. If one digitizes a 3D object even with a dense sampling grid, the reconstructed digital object may have topological distortions and, in general, there exists no upper bound for the Hausdorff distance. This explains why so far no algorithm has been known whic ..."
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Cited by 7 (3 self)
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Digitization is not as easy as it looks. If one digitizes a 3D object even with a dense sampling grid, the reconstructed digital object may have topological distortions and, in general, there exists no upper bound for the Hausdorff distance. This explains why so far no algorithm has been known which guarantees topology preservation. However, as we will show, it is possible to repair the obtained digital image in a locally bounded way so that it is homeomorphic and close to the 3D object. The resulting digital object is always wellcomposed, which has nice implications for a lot of image analysis problems. Moreover, we will show that the surface of the original object is homeomorphic to the result of the marching cubes algorithm. This is really surprising since it means that the wellknown topological problems of the marching cubes reconstruction simply do not occur for digital images of rregular objects. Based on the trilinear interpolation, we also construct a smooth isosurface from the digital image that has the same topology as the original surface. Finally, we give a surprisingly simple topology preserving reconstruction method by using overlapping balls instead of cubical voxels. This is the first approach of digitizing 3D objects which guarantees topology preservation and gives an upper bound for the geometric distortion. Since the output can be chosen as a pure voxel presentation, a union of balls, a reconstruction by trilinear interpolation, a smooth isosurface, or the piecewise linear marching cubes surface, the results are directly applicable to a huge class of image analysis algorithms. Moreover, we show how one can efficiently estimate the volume and the surface area of 3D objects by looking at their digitizations. Measuring volume and surface area of digital objects are important problems in 3D image analysis. Good estimators should be multigrid convergent, i.e., the error goes to zero with increasing sampling density. We will show that every presented reconstruction method can be used for
Linear time recognition algorithms for topological invariants in 3d
 CoRR
"... In this paper, we design linear time algorithms to recognize and determine topological invariants such as genus and homology groups in 3D. These invariants can be used to identify patterns in 3D image recognition and medical image analysis. Our method is based on cubical images with direct adjacency ..."
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Cited by 2 (0 self)
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In this paper, we design linear time algorithms to recognize and determine topological invariants such as genus and homology groups in 3D. These invariants can be used to identify patterns in 3D image recognition and medical image analysis. Our method is based on cubical images with direct adjacency, also called (6,26)connectivity images in discrete geometry. According to the fact that there are only six types of local surface points in 3D and a discrete version of the wellknown GaussBonnett Theorem in differential geometry, we first determine the genus of a closed 2Dconnected component (a closed digital surface). Then, we use the Alexander duality to obtain the homology groups of a 3D object in 3D space. This idea can be extended to general simplicial decomposed manifolds or cell complexes in 3D. 1.
Paths, homotopy and reduction in digital images
 Acta Appl. Math
, 2011
"... Abstract The development of digital imaging (and its subsequent applications) has led to consider and investigate topological notions, welldefined in continuous spaces, but not necessarily in discrete/digital ones. In this article, we focus on the classical notion of path. We establish in particula ..."
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Cited by 2 (2 self)
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Abstract The development of digital imaging (and its subsequent applications) has led to consider and investigate topological notions, welldefined in continuous spaces, but not necessarily in discrete/digital ones. In this article, we focus on the classical notion of path. We establish in particular that the standard definition of path in algebraic topology is coherent w.r.t. the ones (often empirically) used in digital imaging. From this statement, we retrieve, and actually extend, an important result related to homotopytype preservation, namely the equivalence between the fundamental group of a digital space and the group induced by digital paths. Based on this sound definition of paths, we also (re)explore various (and sometimes equivalent) ways to reduce a digital image in a homotopytype preserving fashion.
3D Object Digitization: Majority Interpolation and Marching Cubes
 In Submitted to the International Conference on Image Analysis, ICPR
, 2006
"... In a previous paper [1] we showed that a 3D object can be digitized without changing the topology if the object is rregular and if the reconstruction method fulfills certain requirements. In this paper we give two important examples for such reconstruction methods. First, we introduce Majority Inte ..."
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Cited by 2 (2 self)
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In a previous paper [1] we showed that a 3D object can be digitized without changing the topology if the object is rregular and if the reconstruction method fulfills certain requirements. In this paper we give two important examples for such reconstruction methods. First, we introduce Majority Interpolation, an algorithm to interpolate sampling points at doubled resolution such that topological ambiguities are resolved. Second, we show how the wellknown Marching Cubes algorithm has to be modified such that it is topology preserving. This is the first approach of digitizing 3D objects which guarantees topology preservation for voxelbased or polygonal surfacebased reconstructions. 1
A Solution to the Animal Problem Akira Nakamura 1
, 2010
"... Let A be an animal. We consider magnifications of A in x, y, or zdirection. But this is not always possible because there may be voxels that block the magnification. To overcome this difficulty, we consider an ADmovement. This is performed by considering the magnification in two directions (e.g. ..."
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Let A be an animal. We consider magnifications of A in x, y, or zdirection. But this is not always possible because there may be voxels that block the magnification. To overcome this difficulty, we consider an ADmovement. This is performed by considering the magnification in two directions (e.g., ydirection and zdirection) such that any series of blocking voxels never forms a cycle. If a magnification is possible for two directions then it is also possible for the third direction. After all undilatable voxels have been erased, we can apply a previously shown result for showing that the animal problem is positively solvable. This paper is a continuation of [4]. Let B be a wellcomposed picture [3] that does not contain any cavity or any tunnel. Is B SDequivalent (i.e., equivalent by repeated simple deformations) to a single voxel? This problem was named Bproblem by the late Azriel Rosenfeld. In [4], we have given a positive solution to this problem. The animal problem proposed by Janos Pach was the question whether every animal can be reduced to a single unit cube by a finite sequence of moves, each consisting of either adding or deleting a cube provided the result is an animal again at each move. Azriel Rosenfeld called this problem the Aproblem. In this animal problem, the deformation is not based on SD but must preserve animality of the picture at each step. This means that the Bproblem is weaker than the Aproblem. In this paper, we give a positive solution to the animal problem. We assume that readers are familiar with basic definitions in digital topology, such as provided in [1, 2], and, in particular, with the discussion in [4]. An animal is defined as a topological 3ball in R3, consisting of unit cubes (i.e., a subcomplex in the 3D grid cell space). In accordance with terms of digital topology, hereafter, we call a unit cube also a voxel. Let A be an animal consisting of black voxels (black means that a unit cube is present, i.e., in a given “object”) and p be a black voxel.
Topologypreserving rigid transformation of 2D digital images
, 2013
"... Abstract—We provide conditions under which 2D digital images, considered in the two most common digital topology models (namely, dual adjacency and wellcomposedness), preserve their topological properties under rigid transformation. This study, that is developed in a discrete framework, leads to th ..."
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Abstract—We provide conditions under which 2D digital images, considered in the two most common digital topology models (namely, dual adjacency and wellcomposedness), preserve their topological properties under rigid transformation. This study, that is developed in a discrete framework, leads to the proposal of efficient preprocessing strategies that ensure the topological invariance of images under further rigid transformation. These results and methods are proved to be valid for various kinds of images (binary, greylevel, label), thus providing a generic set of tools, that can be used in particular in the context of image registration and warping. Index Terms—Digital imaging, rigid transformation, digital topology, wellcomposed images, image correction. I.
WELLCOMPOSED IMAGES AND RIGID TRANSFORMATIONS
, 2013
"... We study the conditions under which the topological properties of a 2D wellcomposed binary image are preserved under arbitrary rigid transformations. This work initiates a more global study of digital image topological properties under such transformations, which is a crucial but underconsidered p ..."
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We study the conditions under which the topological properties of a 2D wellcomposed binary image are preserved under arbitrary rigid transformations. This work initiates a more global study of digital image topological properties under such transformations, which is a crucial but underconsidered problem in the context of image processing, e.g., for image registration and warping. Index Terms — Wellcomposed images, rigid transformation, digital topology.